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On 2-superirreducible polynomials over finite fields

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On 2-superirreducible polynomials over finite fields. / Bober, J. W.; Du, Lara; Fretwell, D. et al.
In: Indagationes Mathematicae, Vol. 36, No. 3, 31.05.2025, p. 753-763.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Bober, JW, Du, L, Fretwell, D, Kopp, GS & Wooley, TD 2025, 'On 2-superirreducible polynomials over finite fields', Indagationes Mathematicae, vol. 36, no. 3, pp. 753-763. https://doi.org/10.1016/j.indag.2024.08.005

APA

Bober, J. W., Du, L., Fretwell, D., Kopp, G. S., & Wooley, T. D. (2025). On 2-superirreducible polynomials over finite fields. Indagationes Mathematicae, 36(3), 753-763. Advance online publication. https://doi.org/10.1016/j.indag.2024.08.005

Vancouver

Bober JW, Du L, Fretwell D, Kopp GS, Wooley TD. On 2-superirreducible polynomials over finite fields. Indagationes Mathematicae. 2025 May 31;36(3):753-763. Epub 2025 Apr 24. doi: 10.1016/j.indag.2024.08.005

Author

Bober, J. W. ; Du, Lara ; Fretwell, D. et al. / On 2-superirreducible polynomials over finite fields. In: Indagationes Mathematicae. 2025 ; Vol. 36, No. 3. pp. 753-763.

Bibtex

@article{9790b954842c4da2a4c08e3e4fa992f6,
title = "On 2-superirreducible polynomials over finite fields",
abstract = "We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.",
keywords = "Finite fields, Irreducibility, Polynomial compositions",
author = "Bober, {J. W.} and Lara Du and D. Fretwell and Kopp, {G. S.} and Wooley, {T. D.}",
year = "2025",
month = apr,
day = "24",
doi = "10.1016/j.indag.2024.08.005",
language = "English",
volume = "36",
pages = "753--763",
journal = "Indagationes Mathematicae",
issn = "0019-3577",
number = "3",

}

RIS

TY - JOUR

T1 - On 2-superirreducible polynomials over finite fields

AU - Bober, J. W.

AU - Du, Lara

AU - Fretwell, D.

AU - Kopp, G. S.

AU - Wooley, T. D.

PY - 2025/4/24

Y1 - 2025/4/24

N2 - We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

AB - We investigate k-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most k. Let F be a finite field of characteristic p. We show that no 2-superirreducible polynomials exist in F[t] when p=2 and that no such polynomials of odd degree exist when p is odd. We address the remaining case in which p is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree d. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.

KW - Finite fields

KW - Irreducibility

KW - Polynomial compositions

U2 - 10.1016/j.indag.2024.08.005

DO - 10.1016/j.indag.2024.08.005

M3 - Journal article

AN - SCOPUS:85202776857

VL - 36

SP - 753

EP - 763

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

IS - 3

ER -